Nuprl Lemma : filter-fpf-vals
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[P,Q:A ⟶ 𝔹]. ∀[f:x:A fp-> B[x]].
(filter(λpL.Q[fst(pL)];fpf-vals(eq;P;f)) ~ fpf-vals(eq;λa.((P a) ∧b (Q a));f))
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf: a:A fp-> B[a],
filter: filter(P;l),
deq: EqDecider(T),
band: p ∧b q,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
fpf-vals: fpf-vals(eq;P;f),
fpf: a:A fp-> B[a],
let: let,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
fpf_wf,
bool_wf,
deq_wf,
remove-repeats_wf,
l_member_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
list-subtype,
list_wf,
equal_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
set_wf,
list-cases,
filter_nil_lemma,
zip_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
filter_cons_lemma,
eqtt_to_assert,
map_cons_lemma,
zip_cons_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
functionEquality,
universeEquality,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
productElimination,
setEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P,Q:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:x:A fp-> B[x]].
(filter(\mlambda{}pL.Q[fst(pL)];fpf-vals(eq;P;f)) \msim{} fpf-vals(eq;\mlambda{}a.((P a) \mwedge{}\msubb{} (Q a));f))
Date html generated:
2018_05_21-PM-09_25_55
Last ObjectModification:
2018_02_09-AM-10_21_33
Theory : finite!partial!functions
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